The Finite Volume Method Of Time-Space Fractional Partial Equations

There are many outstanding research achievements in the finite volume method of fractional partial derivative equation,but there are still some problems to be solved,which makes us consider the numerical discretization,stability and convergence analysis of the finite volume method in fractional diffusion equations with constant and variable coefficients in this paper.Firstly,we consider the multi-term time-space fractional diffusion equation.Analytical solution is derived through the Mittagg-Leffler function and the finite difference scheme is obtained by using matrix transfer technology.The maximum node errors and convergence order are calculated in the numerical experiment.Secondly,we study the time-space fractional partial derivative equation with fractional Laplacian operator.We get the fully finite volume discrete scheme by adapting L1-type diecretization with graded temporal meshes and spatial Gauss divergence formula.The numerical discretization is unconditional stable and the convergence order is O(?~min{2-?,r?}+h~2).The spectral decomposition definition is used to obtain numerical scattered schemes of equations with fractional Laplace operator.Numerical results show that our schemes with the temporal graded grid have better convergence performance than that with uniform one.Finally,we consider the time-space fractional diffusion equation on a bounded interval with variable coefficients and two-sided Riemann-Liouville fractional operator in space.Based on the nodal basic functions,the finite volume method is used to separate the fractional equation,and the temporal graded grid for the classical L1-type difference scheme is chosen through an optimal mesh grading 2-?/?.The numerical scheme is conditionally stable and the convergence rate is(O(h~2+-T~?N~min{2-?,r?})?Finally,the numerical experimental results show the correctness of our numerical theoretical analysis.